Born in Bangalore, India in 1929, Shakuntala Devi spent her life as a performer, travelling the world. Her talent? Fast math. From quickly memorizing a shuffled deck of cards onstage with her travelling magician father to finding cube roots of 10-digit numbers within seconds on TV specials, Devi entertained crowds with her marvelous ability to swiftly perform in her head calculations for which most people would require many minutes and a calculator. Yet Devi, who described her ability as "a gift" and "a very automatic reaction," did not attend school as a child; though she longed to study math and other subjects in the classroom, her family was unable to afford the long-term investment of education, and instead she spent her teenage years showcasing her number-crunching talents for an audience.

In this brief piece, Michael Barany, a graduate student in history at Princeton University, offers some perspectives on research he conducted while spending three months in Paris. Barany was examining records of mathematical activity during the 1940s, 1950s, and 1960s. At that time, France was still struggling to rebuild its mathematical activity after the devastation of World War II. In the documents he examined, Barany found much fervent discussion about how to advance mathematics in an increasingly international environment. He also found that the international travels of French mathematicians had a widespread impact, both abroad and at home. "Today's seemingly free flow of mathematical books and papers, as well as the abundance of regional and international meetings and collaborations for producing and sharing new research, are the outcome of years of invisible negotiations amongst individuals and institutions too numerous to name," Barany writes. "To comprehend the modern global mathematician sometimes requires a very local perspective."

Yuri Milner, the Russian entrepreneur and philanthropist, and Mark Zuckerberg of Facebook announced on December 12 that they have established a US$3 million Breakthrough Prize in Mathematics. The first award will be given in 2014. The new prize is in addition to the existing life sciences and physics awards, this year given at a black-tie dinner and ceremony at the Ames Research Center in Mountain View, CA, co-hosted by Vanity Fair, attended by scientists and celebrities and hosted by actor Kevin Spacey. The New York Times reported, "For the new math award, Mr. Milner and Mr. Zuckerberg, in consultation with experts, will choose the first winners. Mr. Milner declined to say how many mathematicians would be chosen, but there could be quite a number of windfalls in store: for the physics prize, there were nine inaugural winners." The Independent article quoted Milner at the ceremony: "Einstein said, 'Pure mathematics is the poetry of logical ideas'. It is in this spirit that Mark and myself are announcing a new Breakthrough Prize in Mathematics. The work that the Prize recognizes could be the foundation for genetic engineering, quantum computing or Artificial Intelligence; but above all, for human knowledge itself."

Friedberg (at left), chair of Boston College's Mathematics Department and a former Los Angeles student, wrote this op-ed piece both in reaction to the recently announced results of the Program for International Student Assessment (in which U.S. students failed to score in the top 20 in any category), and in preparation for the implementation of the Common Core standards. He writes that the keys to improving the nation's students' math abilities are "textbooks, teachers and testing. Getting any of these wrong will leave our young people even further behind." Friedberg notes that most school math texts are "badly flawed," and that we need new, fundamentally different textbooks that promote true understanding. Teachers, he says, "are the heart of the enterprise," but most have learned from poor texts. Twenty years ago Singapore revamped its textbooks so that they were clear and as a result both teachers and students attained higher levels. Finally, regarding testing and especially teaching to the test, he writes that "We should consciously seek the middle ground, valuing good tests but not letting them serve as the sole goal of our education system." This op-ed was reprinted in several publications, including The Gulf Today in the United Arab Emirates. (Photo by Jeff Mozzochi.) See also: Tony Phillips' Take, "After the Pisa report."

This December, the National Museum of Mathematics, or MoMath, in New York City celebrated its first anniversary. As part of the celebration, the museum sponsored an evening event on December 5th entitled "Pythagorizing the Flatiron." During the event, attended by over 2,000 people, some 450 museum members, with glowsticks held end-to-end, were positioned around the iconic Flatiron building--75 on one side, 180 on another side, and 195 on the third side. By using the converse of the Pythagorean Theorem, MoMath was able to show that the triangular footprint of this building is approximately that of a right triangle. In the process, calculations showing that the equation 752 + 1802 = 1952 is true--as well as a few nice geometric proofs of the Pythagorean Theorem--were projected on the sides of the building.

In case you hadn't noticed, the date for the event is significant: the lengths of the sides of the Flatiron Building are proportional to the Pythagorean triple 5-12-13. By sheer coincidence, the date 12/5/13 occurred approximately one week before MoMath's first anniversary. (Photos courtesy of MoMath. See more photos of the event.)

With the year's end approaching, mathematics professor Jason Brown waxes poetic about the utility of mathematical thinking in a variety of career fields and encourages readers to make a New Year's resolution to "add a bit of math to your life." Recounting a recent lecture he delivered to financial analysts, Brown notes that understanding the mathematics behind the cumulative risk generated by repeated low-risk events helped the analysts talk to their clients. Similarly, basic conditional probability can help lawyers accurately compute the odds that an individual fitting the description of a perpetrator actually is the perpetrator, and understanding the "six degrees of separation principle" can help marketing professionals. Brown extols the value of viewing the world through the prism of mathematical thinking, and encourages readers to get some new ideas by picking up a book, attending a lecture, or checking out a page on the internet in the new year.

For those of you who stay up late at night worrying about the continuum hypothesis, relief may be at hand. The continuum hypothesis (or CH, in set-theoretic lingo) holds that there are no infinite sets intermediate in size between the natural numbers and the real numbers; it remains a hypothesis because it cannot be proven (or disproven) using the set of axioms agreed to be the foundation of modern mathematics--ZFC, or "Zermelo-Fraenkel set theory with the axiom of choice." In this comprehensive article, Natalie Wolchover uses the EFI Project--a series of workshops organized by Harvard University set theorist Peter Koellner towards the purpose of Exploring the Frontiers of Incompleteness--as a jumping-off point to survey the cutting edge of research on this most intractable of conjectures. Wolchover delves into two major new theoretical frameworks about the "universe of sets"--each of which rules differently on the continuum hypothesis--and discusses the relevance of questions about "actual infinity" such as the CH for practicing mathematicians.

One major framework goes by the name of "ultimate L," and relies on work published by Hugh Woodin in 2010. In set theory, the simplest model of the universe of sets was developed by Kurt Gödel--the seminal set theorist who proved, among other things, that the CH is compatible with ZFC. Called "L" or the "inner model," Godel's model is built up iteratively from the empty set, and admits no infinite sets of intermediate cardinality between the reals and the natural numbers. Unfortunately, it also does not admit the menagerie of "large cardinals" discovered periodically over the 20th century. While the large cardinals lead to rich mathematical structures with consequences for working mathematicians, they cannot be derived from ZFC, requiring instead a corresponding zoo of "large cardinal axioms." Similarly, each type of large cardinal in the infinite hierarchy of large cardinals seemed to require a unique inner model constructed with a unique toolkit, until Woodin proved in 2010 that an inner model producing the "supercompact" large cardinals would also produce all other large cardinals. Woodin's current work is aimed at constructing this elegant and minimal, but still hypothetical, model of the universe of sets. The other major framework, that of the "forcing axioms", has a different flavor; rather than seeking the smallest, most elegant model of the universe of sets, it seeks the largest and richest possible universe of sets. The technique of "forcing" was developed by seminal set theorist Paul Cohen, who proved that the negation of the CH is compatible with ZFC. According to Stevo Todorcevic, forcing is something like taking the algebraic closure of a field, and the forcing axioms extend the universe of sets in many different directions. An axiom known as "Martin's maximum," discovered in the '80s, extends the universe of sets as far as it can go. In this extended universe, there is a class of real numbers (cardinally) larger than the reals of ZFC, and the CH is thus falsified. Todorcevic and his collaborators have shown that, while inelegant, Martin's maximum has consequences that make many mathematical structures easier to use and understand. Still others--like Stephen Simpson of Penn State University--argue that the concrete, manipulable "actual" infinity of infinite sets (as opposed to the "possible infinity" of limits and the number line) doesn't really exist, and ought to be eliminated from mathematics. Simpson leads an effort to wean mathematics off actual infinity, by showing that the vast majority of theorems can be proved using only the notion of potential infinity. Given this range of perspectives, and the somewhat daunting task of keeping up with them, no mathematician need go without a sound night's sleep.

"Maths Master" columnists Burkard Polster and Marty Ross take aim at the questionable mathematical underpinnings of the Ocean Health Index (OHI), a concept launched in a Nature article in 2012. OHI measures the status of 10 public goals for ocean health—ranging from tourism to biodiversity—in individual countries and the world as a whole, then takes a weighted average of the 10 indicators to arrive at an overall score. As Polster and Ross explain, however, a score that averages values based on such dissimilar concepts is essentially meaningless, as using a single number to represent the status of a goal that depends on both biological factors and human interactions. Using as an example the OHI’s equations for calculating an ocean’s "food provision," Polster and Ross highlight further that the methodology for computing the indicators uses concepts such as logarithms without any apparent need or explanation.

In this brief essay, University of Arizona Regents' Professor Alan C. Newell begins by pointing out the existence of patterns all around us, such as fingerprints, zebra stripes, and the arrangement of seeds at the center of a sunflower. Some people have made a teleological argument for the existence of such patterns, i.e., these patterns serve a purpose. For example, "the seeds on a sunflower head are positioned so that each has the most space possible and therefore best access to nutrients and light." But only recently, Newell notes, have we begun to "understand the hows" of patterns: namely "the physical and chemical processes that lead to pattern formation." For the past decade, Newell's interest in pattern formation has led him and several colleagues to focus on phyllotaxis. In particular, they have studied the spiral arrangement of seeds on a sunflower, which can be counted by numbers belonging to the Fibonacci sequence. He and his colleagues have used their "experimentally informed mathematical model to reproduce the configuration of [a growth-promoting hormone] within a sunflower and, hence, the likely sites for seed initiation." And here, Newell points out, is the "wonderful surprise:"

"The places where our model says the seeds should be are the very same as those positions which an algorithm based on the teleological model would predict. The patterns that nature creates using plain old physics and chemistry are the means by which organisms can pursue optimal strategies. How great is that!"

In this piece taken from his blog in the Business Insider, Andy Kiersz discusses Russell's paradox, and the necessity of careful, axiomatic definitions for avoiding the absurdities that crop up in logic and mathematics otherwise. In "naive set theory," a set is just a collection of things, and if defined correctly, a set can contain itself--one example being the set of all sets. Russell's paradox concerns the set of all sets which do not contain themselves. Is this set a member of itself? If you work your way around the logic, you will see that it simultaneously must be, and cannot be. To get around this paradox and others like it, Kiersz points out that in the set-builder notation of Zermelo-Fraenkel set theory, every set is specified as a subset of a given set of objects. This makes it easy to exclude the word "pizza" and the state of California from the set of things which are not natural numbers--by specifying which things count as, well, things--and effectively prohibits sets from being members of themselves, short-circuiting Russell's paradox.